Abstract
Though the definition of gain from trade extends the definition of social welfare from auctions to markets, from a mathematical point of view the additional dimension added by gain from trade makes it much more difficult to design a gain from trade maximizing mechanism. This paper provides a means of understanding when a market designer can choose the easier path of maximizing social welfare rather than maximizing gain from trade.
We provide and prove the first formula to convert a social welfare approximation bound to a gain from trade approximation bound that maintains the original order of approximation. This makes it possible to compare algorithms that approximate gain from trade with those that approximate social welfare. We evaluate the performance of our formula by using it to convert known social welfare approximation solutions to gain from trade approximation solutions. The performance of all known two-sided markets solutions (that implement truthfulness, IR, BB, and approximate efficiency) are benchmarked by both their theoretical approximation bound and their performance in practice. Surprisingly, we found that some social welfare solutions achieve a better gain from trade than other solutions designed to approximate gain from trade.
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Notes
- 1.
Illustratively, the broad explanation of maximizing social welfare in a two-sided market is that sellers who place a relatively higher cost on a given commodity should end up retaining that commodity while the broad explanation of maximizing gain from trade is that sellers who place a relatively lower cost on a given commodity should end up selling that commodity.
- 2.
- 3.
Maximizes SWF of buying agents and remaining commodities.
- 4.
Note that the requirement for \(W_{ALG}>G_{OPT}+C_{OPT}\) is trivial in the context of two-sided markets where the SWF resulting from unallocated commodities is included as the algorithm can at least gain the SWF resulting from not allocating any commodities.
- 5.
[15]’s theoretical bound is negative unless markets are very large as the bound is not tight and the algorithm only performs well on very large markets.
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Gonen, R., Egri, O. (2019). Two-Sided Markets: Mapping Social Welfare to Gain from Trade. In: Slavkovik, M. (eds) Multi-Agent Systems. EUMAS 2018. Lecture Notes in Computer Science(), vol 11450. Springer, Cham. https://doi.org/10.1007/978-3-030-14174-5_8
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